Brealey, Myers. Principles of Corporate Finance. 7th edition

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Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 6. Making Investment

Decisions with the Net

Present Value Rule

Companies, 2003

146 PART I Value

• The market value of your land would be $100 per acre if you cut and removed the

timber this year. The value of cut-over land is also expected to grow at 4 percent per

year indefinitely.

18. The Borstal Company has to choose between two machines that do the same job but

have different lives. The two machines have the following costs:

Year Machine A Machine B

0 $40,000 $50,000

1 10,000 8,000

2 10,000 8,000

3 10,000  replace 8,000

4 8,000  replace

These costs are expressed in real terms.

a. Suppose you are Borstal’s financial manager. If you had to buy one or the other

machine and rent it to the production manager for that machine’s economic life,

what annual rental payment would you have to charge? Assume a 6 percent real

discount rate and ignore taxes.

b. Which machine should Borstal buy?

c. Usually the rental payments you derived in part (a) are just hypothetical—a way of

calculating and interpreting equivalent annual cost. Suppose you actually do buy

one of the machines and rent it to the production manager. How much would you

actually have to charge in each future year if there is steady 8 percent per year

inflation? (Note: The rental payments calculated in part (a) are real cash flows. You

would have to mark up those payments to cover inflation.)

19. Look again at your calculations for question 18 above. Suppose that technological

change is expected to reduce costs by 10 percent per year. There will be new machines

in year 1 that cost 10 percent less to buy and operate than A and B. In year 2 there will

be a second crop of new machines incorporating a further 10 percent reduction, and so

on. How does this change the equivalent annual costs of machines A and B?

20. The president’s executive jet is not fully utilized. You judge that its use by other officers

would increase direct operating costs by only $20,000 a year and would save $100,000

a year in airline bills. On the other hand, you believe that with the increased use the

company will need to replace the jet at the end of three years rather than four. A new jet

costs $1.1 million and (at its current low rate of use) has a life of six years. Assume that

the company does not pay taxes. All cash flows are forecasted in real terms. The real op-

portunity cost of capital is 8 percent. Should you try to persuade the president to allow

other officers to use the plane?

CHALLENGE

QUESTIONS

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1. One measure of the effective tax rate is the difference between the IRRs of pretax and after-

tax cash flows, divided by the pretax IRR. Consider, for example, an investment I generat-

ing a perpetual stream of pretax cash flows C. The pretax IRR is C/I, and the

after-tax IRR is C(1  T

)/I, where T

is the statutory tax rate. The effective rate, call it T

, is

In this case the effective rate equals the statutory rate.

a. Calculate T

for the guano project in Section 6.2.

b. How does the effective rate depend on the tax depreciation schedule? On the

inflation rate?



C/I  C11  T

2/I

C/I

 T

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 6. Making Investment

Decisions with the Net

Present Value Rule

Companies, 2003

CHAPTER 6 Making Investment Decisions with the Net Present Value Rule 147

c. Consider a project where all of the up-front investment is treated as an expense for tax

purposes. For example, R&D and marketing outlays are always expensed in the U.S.

They create no tax depreciation. What is the effective tax rate for such a project?

2. We warned that equivalent annual costs should be calculated in real terms. We did not

fully explain why. This problem will show you.

Look back to the cash flows for machines A and B (in “Choosing between Long- and

Short-Lived Equipment”). The present values of purchase and operating costs are 28.37

(over three years for A) and 21.00 (over two years for B). The real discount rate is 6 per-

cent, and the inflation rate is 5 percent.

a. Calculate the three- and two-year level nominal annuities which have present values of

28.37 and 21.00. Explain why these annuities are not realistic estimates of equivalent

annual costs. (Hint: In real life machinery rentals increase with inflation.)

b. Suppose the inflation rate increases to 25 percent. The real interest rate stays at 6

percent. Recalculate the level nominal annuities. Note that the ranking of machines

A and B appears to change. Why?

3. Suppose that after-tax investment is defined as the investment outlay minus the present

value of future depreciation tax shields. In the guano project, for example, after-tax in-

vestment would be $10,000,000  2,174,000  $7,826,000. This figure would be entered as

the investment outlay, then future cash flows would be calculated ignoring depreciation.

a. Does this change in format affect bottom-line NPV? Does the format have any

advantages or disadvantages?

b. This format requires discounting depreciation tax shields separately. What should

the discount rate be? Note that depreciation tax shields are safe if the company will

be consistently profitable.

c. If depreciation tax shields are not discounted at the ordinary cost of capital, should

the discount rate for the other cash flows change? Why or why not?

MINI-CASE

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New Economy Transport

The New Economy Transport Company (NETCO) was formed in 1952 to carry cargo and

passengers between ports in the Pacific Northwest. By 2002 its fleet had grown to four ves-

sels, one of which was a small dry-cargo vessel, the Vital Spark.

The Vital Spark is badly in need of an overhaul. Peter Handy, the finance director, has just

been presented with a proposal, which would require the following expenditures:

All estimates of costs and revenues ignore inflation. Mr. Handy’s bankers have suggested that infla-

tion will average 3 percent a year.

Install new engine and associated equipment $185,000

Replace radar and other electronic equipment 50,000

Repairs to hull and superstructure 130,000

Painting and other maintenance 35,000

$400,000

NETCO’s chief engineer, McPhail, estimates the postoverhaul operating costs as follows:

Fuel $ 450,000

Labor and benefits 480,000

Maintenance 141,000

Other 110,000

$1,181,000

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 6. Making Investment

Decisions with the Net

Present Value Rule

Companies, 2003

Fuel $420,000

Labor and benefits 405,000

Maintenance 70,000

Other 90,000

$985,000

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148 PART I Value

The Vital Spark is carried on NETCO’s books at a net value of only $30,000, but could

probably be sold “as is,” along with an extensive inventory of spare parts, for $100,000. The

book value of the spare parts inventory is $40,000.

The chief engineer has also suggested installation of a more modern navigation and con-

trol system, which would cost an extra $200,000.

This additional equipment would not

substantially affect the Vital Spark’s performance, but it would result in the following re-

duced annual fuel, labor, and maintenance costs:

All investments qualify for the seven-year MACRS class.

There is no question that the Vital Spark needs a new engine and general overhaul soon.

However, Mr. Handy feels it unwise to proceed without also considering the purchase of a

new boat. Cohn and Doyle, Inc., a Wisconsin shipyard, has approached NETCO with a new

design incorporating a Kort nozzle, extensively automated navigation and power control

systems, and much more comfortable accommodations for the crew. Estimated annual op-

erating costs of the new boat are

Fuel $370,000

Labor and benefits 330,000

Maintenance 70,000

Other 74,000

$844,000

The crew would require additional training to handle the new boat’s more complex and

sophisticated equipment and this would probably require an expenditure of $50,000 to

$100,000.

The estimated operating costs for the new boat assume that it would be operated in

the same way as the Vital Spark. However, the new boat should be able to handle a larger

load on some routes, and this might generate additional revenues, net of additional out-

of-pocket costs, of as much as $100,000 per year. Moreover, a new boat would have a use-

ful service life of 20 years or more. The Vital Spark, even if rehabilitated, could not last

that long—probably only 15 years. At that point it would be worth only its scrap value

of about $40,000.

Cohn and Doyle offered the new boat for a fixed price of $2,000,000, payable half im-

mediately and half on delivery in nine months. Of this amount $600,000 was for the en-

gine and associated equipment and $510,000 was for navigation, control, and other elec-

tronic equipment.

NETCO was a private company, soundly financed and consistently profitable. Cash on

hand was sufficient to rehabilitate or improve the Vital Spark but not to buy the new boat.

However, Mr. Handy was confident that the new boat could be financed with medium-term

debt, privately placed with an insurance company. NETCO had borrowed via a private

placement once before when it negotiated a fixed rate of 12.5 percent on a seven-year loan.

Preliminary discussions with NETCO’s bankers led Mr. Handy to believe that the firm could

arrange an 8 percent fixed-rate medium-term loan.

NETCO had traditionally estimated its opportunity cost of capital for major business

investments by adding a risk premium of 10 percentage points to yields on newly issued

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 6. Making Investment

Decisions with the Net

Present Value Rule

Companies, 2003

Visit us at www.mhhe.com/bm7e

CHAPTER 6 Making Investment Decisions with the Net Present Value Rule 149

Treasury bonds.

Mr. Handy thought this was a reasonable rule of thumb for the dry-

cargo business.

Questions

1. Calculate equivalent annual costs of the three alternatives—overhaul, overhaul with

improved navigation and control, or a brand-new boat. To do the calculation, you will

have to prepare a spreadsheet table showing all costs after taxes over each investment’s

economic life. Take special care with your assumptions about depreciation tax shields

and inflation.

In 2002 Treasury bonds were yielding 5 percent.

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

I. Value 6. Making Investment

Decisions with the Net

Present Value Rule

Companies, 2003

RELATED WEBSITES

PART 1

WEBSITES

Chapter 1 described the role of the financial man-

ager. More information on careers in finance can

be found at:

www

.careers-in-finance.com

The following websites, which are concerned

largely with personal finance, provide discus-

sions of the time value of money and calculators:

www

.bankrate.com

www

.financenter.com

www

.financialplayerscenter.com

www.invest-faq.com

www.money.cnn.com

www.unb.ca/web/transpo/mynet/

mtw21b.htm (how to use Excel for

compound interest calculations)

One of the few sites with material on capital in-

vestment decisions is:

www

.asbdc.ualr.edu/fod/1518.htm

Chapter 3 explained how bonds are valued. Help-

ful material and data on bond markets are avail-

able on:

www

.bondsonline.com (good bond data)

http://bonds.yahoo.com

www.finpipe.com (good explanations of bond

markets)

www

.fintools.net (contains a bond calculator)

www.ganesha.org (explanation of bond

markets and calculator)

www

.hsh.com (good bond data)

www.investinginbonds.com (also contains

links to related sites)

www

.investorguide.com/university.html

(good explanations of bond and equity

markets)

http:

//money.cnn.com/markets/bondcenter

Chapter 4 was concerned with stock markets and

equity valuation. Most major stock exchanges

have good websites. See, for example:

www

.nyse.com

(New York Exchange)

www

.nasdaq.com

(NASDAQ)

www

.londonstockexchange.com

(London

Exchange)

www

.tse.or.jp

(Tokyo Exchange)

www

.123world.com/stockexchanges

(links

to exchanges)

www

.fibv.com

(The World Federation of

Exchanges publishes useful comparative

statistics)

Data on stock market indexes can be found on:

www

.djindexes.com (Dow Jones index)

www.spglobal.com (Standard & Poor’s

indexes)

www

.barra.com (market indices with

information on dividend yields, P/Es etc.)

www

.wilshire.com

There is a large number of sites with market com-

mentary and data on individual firms and stocks.

We find Finance.Yahoo particularly useful.

http:

//finance.yahoo.com

www

.bloomberg.com

http:

//hoovers.com

www

.cbs.marketwatch.com

www

.finance.lycos.com

http://money.cnn.com

http://moneycentral.msn.com

www.wsrn.com

http://my.zacks.com (includes earnings

forecasts)

The following sites provide useful software and

data for calculating company values:

http:

//financialplayerscenter.com

www.valuepro.net

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

II. Risk 7. Introduction to Risk,

Return, and the Opportunity

Cost of Capital

Companies, 2003

CHAPTER SEVEN

152

INTRODUCTION TO

RISK, RETURN, AND

THE OPPORTUNITY

COST OF CAPITAL

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

II. Risk 7. Introduction to Risk,

Return, and the Opportunity

Cost of Capital

Companies, 2003

WE HAVE MANAGED to go through six chapters without directly addressing the problem of risk, but

now the jig is up. We can no longer be satisfied with vague statements like “The opportunity cost of

capital depends on the risk of the project.” We need to know how risk is defined, what the links are

between risk and the opportunity cost of capital, and how the financial manager can cope with risk

in practical situations.

In this chapter we concentrate on the first of these issues and leave the other two to Chapters 8

and 9. We start by summarizing 75 years of evidence on rates of return in capital markets. Then we

take a first look at investment risks and show how they can be reduced by portfolio diversification.

We introduce you to beta, the standard risk measure for individual securities.

The themes of this chapter, then, are portfolio risk, security risk, and diversification. For the most

part, we take the view of the individual investor. But at the end of the chapter we turn the problem

around and ask whether diversification makes sense as a corporate objective.

153

Financial analysts are blessed with an enormous quantity of data on security prices

and returns. For example, the University of Chicago’s Center for Research in Secu-

rity Prices (CRSP) has developed a file of prices and dividends for each month since

1926 for every stock that has been listed on the New York Stock Exchange (NYSE).

Other files give data for stocks that are traded on the American Stock Exchange and

the over-the-counter market, data for bonds, for options, and so on. But this is sup-

posed to be one easy lesson. We, therefore, concentrate on a study by Ibbotson As-

sociates that measures the historical performance of five portfolios of securities:

1. A portfolio of Treasury bills, i.e., United States government debt securities

maturing in less than one year.

2. A portfolio of long-term United States government bonds.

3. A portfolio of long-term corporate bonds.

4. Standard and Poor’s Composite Index (S&P 500), which represents a

portfolio of common stocks of 500 large firms. (Although only a small

proportion of the 7,000 or so publicly traded companies are included in the

S&P 500, these companies account for over 70 percent of the value of stocks

traded.)

5. A portfolio of the common stocks of small firms.

These investments offer different degrees of risk. Treasury bills are about as safe

an investment as you can make. There is no risk of default, and their short maturity

means that the prices of Treasury bills are relatively stable. In fact, an investor who

wishes to lend money for, say, three months can achieve a perfectly certain payoff

by purchasing a Treasury bill maturing in three months. However, the investor can-

not lock in a real rate of return: There is still some uncertainty about inflation.

By switching to long-term government bonds, the investor acquires an asset

whose price fluctuates as interest rates vary. (Bond prices fall when interest rates

rise and rise when interest rates fall.) An investor who shifts from government to

7.1 SEVENTY-FIVE YEARS OF CAPITAL MARKET

HISTORY IN ONE EASY LESSON

The two bond portfolios were revised each year to maintain a constant maturity.

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

II. Risk 7. Introduction to Risk,

Return, and the Opportunity

Cost of Capital

Companies, 2003

corporate bonds accepts an additional default risk. An investor who shifts from cor-

porate bonds to common stocks has a direct share in the risks of the enterprise.

Figure 7.1 shows how your money would have grown if you had invested $1 at

the start of 1926 and reinvested all dividend or interest income in each of the five

portfolios.

Figure 7.2 is identical except that it depicts the growth in the real value

of the portfolio. We will focus here on nominal values.

Portfolio performance coincides with our intuitive risk ranking. A dollar invested

in the safest investment, Treasury bills, would have grown to just over $16 by 2000,

barely enough to keep up with inflation. An investment in long-term Treasury bonds

would have produced $49, and corporate bonds a pinch more. Common stocks were

in a class by themselves. An investor who placed a dollar in the stocks of large U.S.

firms would have received $2,587. The jackpot, however, went to investors in stocks

of small firms, who walked away with $6,402 for each dollar invested.

Ibbotson Associates also calculated the rate of return from these portfolios for

each year from 1926 to 2000. This rate of return reflects both cash receipts—

dividends or interest—and the capital gains or losses realized during the year.

Averages of the 75 annual rates of return for each portfolio are shown in Table 7.1.

154 PART II

Risk

1926 1936 1946 1956 1966 1976 1986 2000

100

1,000

10,000

Dollars

Year

6,402.2

2,586.5

64.1

48.9

16.6

Small firms

S&P 500

Corporate bonds

Government bonds

Treasury bills

FIGURE 7.1

How an investment of $1 at the start of 1926 would have grown, assuming reinvestment of all dividend and interest

payments.

Source: Ibbotson Associates, Inc., Stocks, Bonds, Bills, and Inflation, 2001 Yearbook, Chicago, 2001; cited hereafter in this chapter as

Portfolio values are plotted on a log scale. If they were not, the ending values for the two common stock

portfolios would run off the top of the page.

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

II. Risk 7. Introduction to Risk,

Return, and the Opportunity

Cost of Capital

Companies, 2003

Since 1926 Treasury bills have provided the lowest average return—3.9 percent

per year in nominal terms and .8 percent in real terms. In other words, the average

rate of inflation over this period was just over 3 percent per year. Common stocks

were again the winners. Stocks of major corporations provided on average a risk

premium of 9.1 percent a year over the return on Treasury bills. Stocks of small firms

offered an even higher premium.

You may ask why we look back over such a long period to measure average rates

of return. The reason is that annual rates of return for common stocks fluctuate so

CHAPTER 7

Introduction to Risk, Return, and the Opportunity Cost of Capital 155

1926 1936 1946 1956 1966 1976 1986 2000

100

1,000

10,000

Dollars

Year

659.6

266.5

6.6

5.0

1.7

Small firms

S&P 500

Corporate bonds

Government bonds

Treasury bills

FIGURE 7.2

How an investment of $1 at the start of 1926 would have grown in real terms, assuming reinvestment of all

dividend and interest payments. Compare this plot to Figure 7.1, and note how inflation has eroded the purchasing

power of returns to investors.

Average Annual

Average Risk Premium

Rate of Return

(Extra Return Versus

Portfolio Nominal Real Treasury Bills)

Treasury bills 3.9 .8 0

Government bonds 5.7 2.7 1.8

Corporate bonds 6.0 3.0 2.1

Common stocks (S&P 500) 13.0 9.7 9.1

Small-firm common stocks 17.3 13.8 13.4

TABLE 7.1

Average rates of return on

Treasury bills, government

bonds, corporate bonds,

and common stocks,

1926–2000 (figures in

percent per year).

Source: Ibbotson Associates,

Inc., 2001 Yearbook.

Brealey−Meyers:

Principles of Corporate

Finance, Seventh Edition

II. Risk 7. Introduction to Risk,

Return, and the Opportunity

Cost of Capital

Companies, 2003

much that averages taken over short periods are meaningless. Our only hope of gain-

ing insights from historical rates of return is to look at a very long period.

Arithmetic Averages and Compound Annual Returns

Notice that the average returns shown in Table 7.1 are arithmetic averages. In

other words, Ibbotson Associates simply added the 75 annual returns and di-

vided by 75. The arithmetic average is higher than the compound annual return

over the period. The 75-year compound annual return for the S&P index was

11.0 percent.

The proper uses of arithmetic and compound rates of return from past investments

are often misunderstood. Therefore, we call a brief time-out for a clarifying example.

Suppose that the price of Big Oil’s common stock is $100. There is an equal

chance that at the end of the year the stock will be worth $90, $110, or $130. There-

fore, the return could be ⫺10 percent, ⫹10 percent, or ⫹30 percent (we assume

that Big Oil does not pay a dividend). The expected return is

⁄3(⫺10 ⫹10 ⫹30)

⫽⫹10 percent.

If we run the process in reverse and discount the expected cash flow by the ex-

pected rate of return, we obtain the value of Big Oil’s stock:

The expected return of 10 percent is therefore the correct rate at which to discount

the expected cash flow from Big Oil’s stock. It is also the opportunity cost of capi-

tal for investments that have the same degree of risk as Big Oil.

Now suppose that we observe the returns on Big Oil stock over a large number

of years. If the odds are unchanged, the return will be ⫺10 percent in a third of the

years, ⫹10 percent in a further third, and ⫹30 percent in the remaining years. The

arithmetic average of these yearly returns is

Thus the arithmetic average of the returns correctly measures the opportunity cost

of capital for investments of similar risk to Big Oil stock.

The average compound annual return on Big Oil stock would be

1.9 ⫻ 1.1 ⫻ 1.32

冫

⫺ 1 ⫽ .088, or 8.8%,

⫺10 ⫹ 10 ⫹ 30

⫽⫹10%

PV ⫽

110

1.10

⫽ $100

156 PART II

Risk

We cannot be sure that this period is truly representative and that the average is not distorted by a few

unusually high or low returns. The reliability of an estimate of the average is usually measured by its

standard error. For example, the standard error of our estimate of the average risk premium on common

stocks is 2.3 percent. There is a 95 percent chance that the true average is within plus or minus 2 stan-

dard errors of the 9.1 percent estimate. In other words, if you said that the true average was between

4.5 and 13.7 percent, you would have a 95 percent chance of being right. (Technical note: The standard

error of the average is equal to the standard deviation divided by the square root of the number of ob-

servations. In our case the standard deviation is 20.2 percent, and therefore the standard error is

)

This was calculated from (1 ⫹ r)

⫽ 2,586.5, which implies r ⫽ .11. Technical note: For lognormally dis-

tributed returns the annual compound return is equal to the arithmetic average return minus half the

variance. For example, the annual standard deviation of returns on the U.S. market was about .20, or 20

percent. Variance was therefore .20

, or .04. The compound annual return is .04/2 ⫽ .02, or 2 percent-

age points less than the arithmetic average.

20.2

冫

275 ⫽ 2.3.